Asymptotic expansion of solutions to nonlinear elliptic eigenvalue problems

Abstract

We consider the nonlinear eigenvalue problem -Δu + g(u) = λ sin u in Ω, it > 0 in Ω, it = 0 on ∂Ω, where Ω C R N (N > 2) is an appropriately smooth bounded domain and A > 0 is a parameter. It is known that if λ » 1, then the corresponding solution u λ is almost flat and almost equal to π inside Ω. We establish an asymptotic expansion of u λ (x) (x ∈ Ω) when A » 1, which is explicitly represented by g.